melissa soenke march 11, 2014 - university of arizona
TRANSCRIPT
EEG data from one time point can be conceptualized as a line traveling through high dimensional space
Revisiting Chapter 10 and the Dot Product “A vector with two elements can be conceptualized as
a point in 2-D space. A vector with three elements can therefore be conceptualized as a point in a 3-D space (a cube), and so on for as many dimensions as there are elements in the vector.”
In PCA each dimension corresponds to each electrode
“The goal of PCA is to construct sets of weights (called principal components) based on the covariance of a set of correlated variables (electrodes) so that components explain all the variance of the data.”
Uncorrelated with each other
Created so the first component explains as much variance as possible for one electrode, the second as much of the residual variance as possible for one electrode while still being orthogonal to the first, and so on for as many components as there are electrodes
Geometric – each component is a vector in a space characterized by as many dimensions as there are electrodes that characterizes the direction of the data distribution
As a set of spatial filters Weights for each electrode are defined by patterns of
interelectrode temporal covariance
SURFACE LAPLACIAN PCA
Weights are defined only by statistical properties, not physical location on the scalp
Identifies large-scale covariance and highlights global spatial features of the data
Weights are defined by interelectrode distance
Attenuates low-spatial frequency activity and highlights local features of the data
As a data reduction technique High dimensional data (like our 64 electrode data)
can be reduced to a smaller number of dimensions Components that account for a large amount of
variance reflect true signal Components that account for less variance reflect
noise
Step 1: construct a covariance matrix
Data can be organized in 3 ways:1. PCA can be computed from the ERP2. PCA can be computed from all time points from all
trials3. PCA can be computed separately for each trial
Covariance = (n-1)-1 (X-X) (X- X)T
Covariance of ERP
P1 F4 P8 P1 F4 P8
P1
F4
P8
P1
F4
P8
Average covariance of single-trial EEG
P1 F4 P8 P1 F4 P8
P1
F4
P8
P1
F4
P8
Covariance of single-trial EEG
P1 F4 P8 P1 F4 P8
P1
F4
P8
P1
F4
P8
Reflects phase-locked covariance
Reflects total (phase-locked & non-phase-locked) covariance
Step 2: Perform an eigendecomposition
Eigendecomposition is a matrix decomposition that returns eigenvectors and associated eigenvalues which characterize the patterns of interelectrode covariance
Eigenvectors and values exist in pairs An eigenvector is a direction An eigenvalue is a number, telling you how
much variance there is in the data in that direction
Let’s take a look at this visually using the following address:
http://georgemdallas.wordpress.com/2013/10/30/principal-component-analysis-4-dummies-eigenvectors-eigenvalues-and-dimension-reduction/
In PCA, the eigenvalues are the principal components or weights for each electrode that, applied to the electrode time series, produce the PCA time courses
Eigenvalues can be scaled to percentages of variance accounted for – divide each eigenvalue by the sum of all the eigenvaluesand multiply by 100
Results are square matrices with as many rows/columns as electrodes
Electrode weights for each component can be plotted as topographical maps and time courses of components can be obtained by multiplying weights of electrodes by electrode time-series data
FOR ERP FOR SINGLE TRIAL
PC #1, eigval=67.3325 PC #2, eigval=12.4658 PC #3, eigval=11.5038
PC #4, eigval=2.1677 PC #5, eigval=1.6782 PC #6, eigval=1.1696
PC #7, eigval=0.6907 PC #8, eigval=0.52854 PC #9, eigval=0.4413
PC #1, eigval=60.4833 PC #2, eigval=11.0014 PC #3, eigval=4.4183
PC #4, eigval=3.4652 PC #5, eigval=2.5943 PC #6, eigval=1.646
PC #7, eigval=1.286 PC #8, eigval=1.2426 PC #9, eigval=1.0859
To determine which components are significant you need to establish a percentage variance threshold. Components above this are significant.
1. Compute % explained variance expected from each component if all electrodes are uncorrelated with each other
2. Permutation testing –data are randomly shuffled and PCA is computed on shuffled data, then the amount of explained variance, averaged over repetitions, is taken as threshold
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% var. accounted forchance-level (alg)chance-level (perm.test)
With significant components you have many analysis options
1. You can use PCA as a dimension reduction technique and analyze only significant components
2. You can examine how much variance is accounted for by the signal components
3. You can compute the variance accounted for by all the nonsignificant components to estimate noise
4. You can learn information about the complexity of a system
PCA forces components to be orthogonal
Components can be rotated to allow them to be correlated or to allow them to capture additional variance
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Time (ms)
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PC1 from -100 ms PC1 from 200 ms PC1 from 500 ms PC1 from 1000 ms
“PCA can be combined with temporal bandpass filtering to highlight frequency-band-specific spatial features.”
First apply bandpass filter to time-domain signal, then perform PCA
FOR SINGLE TRIALALL FREQUENCIES
FOR SINGLE TRIAL12 HZ FILTERED
PC #1, eigval=55.0566 PC #2, eigval=17.7757 PC #3, eigval=10.6846
PC #4, eigval=3.6782 PC #5, eigval=2.3338 PC #6, eigval=2.1297
PC #7, eigval=0.92124 PC #8, eigval=0.81833 PC #9, eigval=0.69146
PC #1, eigval=60.4833 PC #2, eigval=11.0014 PC #3, eigval=4.4183
PC #4, eigval=3.4652 PC #5, eigval=2.5943 PC #6, eigval=1.646
PC #7, eigval=1.286 PC #8, eigval=1.2426 PC #9, eigval=1.0859
PCA can be used to test condition differences PCA can be computed for all conditions
Advantage 1: signal-to-noise ratio of covariance is maximized because many trials contribute to matrix
Advantage 2: facilitates interpretation of condition differences because differences can’t be due to topographical differences in PC weights
PCA can be computed for each condition separately Advantage: increased sensitivity to identifying
condition differences because weights are tuned to each condition